http://www.slideshare.net/aranglenn/automatic-car-parking http://www.sciencedirect.com/science/article/pii/S1110866511000077 http://singularityhub.com/2009/10/28/self-parking-car-from-stanford-and- volkswagen/ Egyptian Informatics Journal Volume 12, Issue 1 , March 2011, Pages 9–17 Original article An intelligent hybrid scheme for optimizing parking space: A Tabu metaphor and rough set based approach Soumya Banerjee a , , , Hameed Al-Qaheri b , a Birla Institute of Technology, CS Dept., Mesra, India b Kuwait University CBA, QMIS Dept., Kuwait Received 6 June 2010. Accepted November 2010. Available online 24 March 2011. http://dx.doi.org/10.1016/j.eij.2011.02.006 , How to Cite or Link Using DOI Cited by in Scopus (0) Permissions & Reprints Abstract Congested roads, high traffic, and parking problems are major concerns for any modern city planning. Congestion of on-street spaces in official neighborhoods may give rise to inappropriate parking areas in office and shopping mall complex during the peak time of official transactions. This paper proposes an intelligent and optimized scheme to solve parking space problem for a small city (e.g., Mauritius) using a reactive search technique (named as Tabu Search) assisted by rough set. Rough set is being used for the extraction of
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Figure 2. Proposed polytonic positions of parking spaces A, B and C – with different edge length.
Similarly, once the different positing of parking spaces have been identified, the other toll
rough set is introduced to improvise the marginal parking scenario and thus lead toward
better optimization.
There are two general kinds of decisional rules in classic rough set theory [25]. The first is
the exact decisional rule, named also deterministic, where the decisional set (the cost)
contains the conditional attributes (area or other features). The second is the approximate
decisional rule in which only some conditional attributes (area or other features) are
included in the decisional set (price) [5] and [26]. In this case of parking function, the causal
relationships between the property features and its value are appraised without any
uncertainty. The logical prepositions if…then allow the user to create a preferential system
based on the property market data. The granularity of the system, its uncertainty can be
increased in case the information is based on a few observations [5]. An example [26] of
rule that may be used for valuation purposes is indicated below:
If near_ parking_dist = x ∧ time_occupancy = “LESS” ∧ Date ∧ YEARS = Z → cost of parking
is high) //Sample Parking Map Rule 1
If arrival_time = y (not peak hour), then y ∨ Z ∧ probability of available parking space → cost of parking is less with high probability of parking space
//Sample Parking Map Rule 2
Let A = (U, IND (B)) be an approximation space. A pair (L,U) ∈ P(U) × P(U) is called a rough
set in A, whereL = A(X), U = A(X) for some X ⊆ U and P(U) is the power set of U. Due to the
granularity of knowledge, rough sets cannot be characterized by using available knowledge.
Therefore, for every rough set X, we associate two crisp sets, called lower and upper
approximation. Intuitively, the lower approximation of X consists of all elements that surely
belong to X, the upper approximation of X consists of all elements that possibly belong to X,
and the boundary region of X consists of all elements that cannot be classified uniquely to
the set or its complement, by employing the available knowledge [33]. In the following, the
measure proposed for the parking information system has been simplified,
where U = {u1, u2, u24} and A = {a1; a2; a3; a4 witha1 = effective parking slot, a2 = average
numbers of vehicle to be parked on a day, a3 = duration anda4 = different parking
probabilities for drivers. Let R be an equivalence relation defined on U and let R1 = {a1,a2}
and R2 = {a1; a2; a3} [1].
Let S = (U, A) be an information system and U = A = {X, X2, …, Xm}
X6 {u25, u26, u27, u28,, u31} Marginal resultant of R1X = R2X With overlaps
This is caused by the fact that R1 and R2 have different knowledge granulations. Thus, under
the measure proposed in this paper, the uncertainties of X with respect to different
equivalence relations are well characterized [1].
Rough set system helps us determine in real time for every parking tariff class, the time
moment after which there are no longer vacant parking spaces for the drivers requesting it.
In any time moment t, we will know the remaining time interval T(t) for selling parking spaces
to a particular parking tariff class. It is clear that T(t)depends on the cumulative numbers of
drivers requesting a particular tariff class Di (t), (i = 1, 2, 3, ...). The cumulative numbers of
driver requests are antecedents, while remaining time period for selling the parking
spaces T(t) is the consequence [2] and [27]. As such a parking information system is a
pair S = (U; A), where,
•
U is a non-empty finite set of objects which is the number of cars C here to be under
queue of parking; nare the parking slots and k are un-parked drivers if any.
•
For every a ∈ A, there is a mapping a, a: U → Va, where Va is called the value set of a.
Each subset of attributes P ⊆ A determines a binary indistinguishable
relation Opti(Park) as follows:
(1)
Opti(Park)={(u,v)∈U×U|∀a∈P,a(u)=a(v)}
It can be easily shown that Opti(Park) is an equivalence relation on the set U[28]. Similarly,
to formulate theμ, this is the valid move for parking car, we can define a decision table as an
information system S=(U,C∪D)withC∩D=∅, where, an element of C is called a condition
attribute, an element of D is called a decision attribute [1]. Table 1 demonstrates the
parking area assigned in a small city e.g. Mauritius and their actual capacity defined in
terms of parking space. Depending on the data provided in Table 1 and Table 2present the
effective and functional parking slots available.
•
Statistics of car parking (mostly taxi) of seven continuous days have been recorded
(Table 2).
•
Day 5th and 6th (Bold part in Table 2) envisage high density of car arrived and least
parking spaces available, respectively.
•
The duration of parking varies from minimum 15 min to 5 h in a particular day.
•
Statistically, it is also observed that for each day the parking space is not adequate
except the day 5th when only 31 cars did not get appropriate available parking space
in the city.
•
The statistical data are used for the proposed simulation for finding the optimal space
of through intelligent methodology.
Table 1. Parking slot distribution in Mauritius.
Name of parking area
Actual parking slots available
Port Louis 1598
Curepipe 170
Rose Hill 230
Quatre Bornes 127
Data provided by Department of Public Infrastructure, Land and Transport, Govt. of Mauritius.
Table 2. Statistical snapshot about car parking in Mauritius (weekly data).
Day
Effective parking slots available Avg. No. of vehicles on a day No. vehicle parked Duration (min./max.)
1 1590 988 923 30 min/4 h
2 1582 1005 910 15 min/3 h
3 1587 990 934 30 min/4 h
4 1584 885 780 45 min/3 h
5 1591 1021 990 311 min/5 h
6 1576 940 880 15 min/3 h
7 1588 970 905 30 min/4 h
The available number of parking spaces must be updated every time a driver is accepted for
parking. The algorithm describes the rough set guided Tabu Search engine for parking space
optimization and control in the following steps:
Step 1: Record all cumulative Ri (t) that are based on a large number of driver requests for
parking in the parking slot in a particular area. Similarly, establish an objective function
based on the constraints.
Step 2: Analyze the area, where the parking slot is located and the following parameters are
recorded:
•
the asymptotical behavior of the parking slot distribution;
•
limiting probability that all parking spaces are occupied;
•
let a(r,s,k) denotes the numbers of choices for which r spaces remain
unoccupied, s spaces are occupied at the end, and k people drive home [20];
•
this also leads to the fact that there are n = r + s spaces in total, and
that m = k + s drivers arrive [20].
•
Hence, Opti(Park) is recursive in nature and may be expressed as
follows [2] and [20]:
(2)
Note: This will not be true when the uncertain distribution is involved.
Step 3: Formulate a corresponding optimization problem [according
to Opti(Park)={(u,v)∈U×U|∀a∈P,a(u)=a(v)}] and find the optimal parking move μ, for each
generated drive’s assignment for parking comprising of asgd (n, m, k).
Step 4: Based on the statistical data resulting from Steps 1 and 2, use some of the existing
algorithms [1] to generate the rough set as upper and lower approximation.
3.2. Stopping criteria and measurement of optimal parking
Various stopping conditions can be applied to the proposed algorithm. The algorithm may
stop when a maximum number of evaluations; a minimum value of the weighted variance or
a maximum number of iterations (without any significant improvement in the objective
function) is reached. During any iteration for searching optimal parking space, the length of
the Tabu list is calculated as half of the smallest distance between the Tabu points and the
marginal points and could be represented as [24]:
(3)
where subscripts u and t represent the promising points and the Tabu lists, respectively. The parameter vtspecifies the minimum Euclidean distance that must exist between each
random point and Tabu points. It is important to note that using this scheme, the size of the Tabu list dynamically changes based on the distribution of the uncertain and the Tabu points in the parking solution space [24].
Thus, Eq. (3) is a measurable point to analyze the proposed algorithm and differs
from [24] by a coefficient multiplication factor uk(0-1), which considers all the marginal and
near conditions evolved in real parking scenario. The range of this varies from 0 to 1 units of
near ness.
4. Result and implicationsThe proposed model rough set guided Tabu hybrid metaphor has been implemented on a
test data using within MS Windows environment using C++. The simulation is supported by
3D plot demonstrating as followings:
•
initial parking space 1 against gradient and number of Tabu iterations;
•
parking space 2 against gradient and number of Tabu iterations;
•
identifying extreme parking space, when the uncertainty distribution in parking
function becomes high.
The three parameter sets help to investigate the performance of the proposed algorithm.
In the post implementation phase of the proposed algorithm, a 3D plot is prepared in Fig. 3,
which comprises a number of iterations. This plot demonstrates availability and probability
of parking space in the first set of iterations. The gradient value is considered under rough
set rule. It implies that, in the first set of search optimization, parking space becomes stiffer
and most of the cars were un-parked (similar to the statistics of 4th day given in Table
2 where 105 cars remained un-parked). Only gray colored portion in the plot was facilitated
(0.2–0.3 units of parking space 1) with proper parking space. Similarly, the value of gradient
becomes stiffer but covers wider parking space. Still the convergence is not able to cover
the uncertain and extreme conditions. Therefore, plot of parking space 2 against gradient
and Tabu iterations shown in Fig. 4offers >1.2 units of space in parking slot. This case is
analogous to Cayleey’s formula and there are nn−2labeled trees on n vertices, where car
number 9 is just entering into parking queue expecting car number 6 is coming out of the
queue (Fig. 1 in Section 3.2). Hence, it is an example of marginal parking condition.
Figure 3. Initial parking space and gradient value (under rough set).
Figure 4. Intermediate parking space and gradient value – marginal parking.
In Fig. 5 extreme car parking criterion is presented through proposed rough set guided Tabu
Search. The performance is satisfactory and covers more confident parking space till 0.4 unit
space and still maintains a straight gradient irrespective of uncertain conditions. The